I can see one of two things happening:
- The drive belt is worn out.
- The tensioner pulley which takes up the slack in the drive belt is not doing its job. If this is the case, the spring associated with the tensioner pulley may be weak.
The first one may be easy enough. If this has not been changed out in several years, you may want to just do this in the first place. It should be fairly easy to replace, but there will be a few guides which will need to be removed to get the belt off/on. The belt itself is most likely made of rubber with some type of cloth (sort of) which is wound around the outside of it to provide grip on the pulleys. If the outer layer is worn off on the faces which comes in contact with the pulleys, this would be an obvious cause.
The second one may not be as easy to diagnose. The pulley in question should be able to be identified when you press on the clutch. The tensioner pulley should be the pulley which moves to release tension on the belt. Obviously this is going to be a two person job, or you're going to have to be really flexible to figure it out. This would most easily be accomplished with the lawn mower up in the air, but that's easier said than done. Be that as it may, when you find the tensioner pulley, see if you can apply more pressure to the pulley (same direction of travel as when the pulley is under tension). If there is a bunch of slack there, the spring which holds the tension may have bit the dust or may be stretched so as to not provide the amount of tension needed to do the task.
There is a third thing which I can think of, that being both the tensioner mechanism and the belt are worn out. I'd start with the belt, though, as that is an obvious wear item.
If you go to this page on the Sears site, then click on "Ground Drive", it will bring up a diagram. The two pulleys I see which could be what you are looking for is either part #38 or #47. More than likely it's #47. I'm not sure how the mechanism works, but if #47 is it, whatever action is taken with a foot clutch or gear engagement will cause this too move. This will then take up the slack to cause the belt to catch. The entire "clutch system" of the mower is based on the friction of the belt against the pulleys. If the friction is gone, the mower won't move.